The speed of the current of a stream is 1 km/h. A motor boat goes 35 km upstream and then back 35 km downstream to its starting point in a total of 12 hours. What is the speed of the motor boat in still water?

Introduction / Context:
This boats and streams problem deals with a round trip: the boat travels upstream a certain distance and then returns downstream to the starting point. You are given the speed of the current and the total time for the entire trip, and asked to find the speed of the boat in still water. Such questions strengthen understanding of relative speed concepts and equation formation.

Given Data / Assumptions:

• Distance upstream = 35 km; distance downstream back = 35 km.
• Speed of current c = 1 km/h.
• Let speed of boat in still water be b km/h.
• Upstream speed = b − 1 km/h.
• Downstream speed = b + 1 km/h.
• Total time for upstream and downstream together = 12 hours.

Concept / Approach:
We use time = distance / speed for each leg of the journey and add the times. This gives an equation involving b, which we then solve. The main idea is that the same distance is traveled twice, but at different effective speeds due to the current, giving different times.

Step-by-Step Solution:
Step 1: Time taken to go upstream 35 km = 35 / (b − 1) hours.Step 2: Time taken to return downstream 35 km = 35 / (b + 1) hours.Step 3: Total time is given as 12 hours, so we write: 35 / (b − 1) + 35 / (b + 1) = 12.Step 4: Factor 35: 35[1/(b − 1) + 1/(b + 1)] = 12.Step 5: Combine the fractions inside brackets: 1/(b − 1) + 1/(b + 1) = (b + 1 + b − 1) / (b^2 − 1) = 2b / (b^2 − 1).Step 6: So the equation becomes 35 * (2b / (b^2 − 1)) = 12.Step 7: Simplify: 70b / (b^2 − 1) = 12.Step 8: Cross multiply: 70b = 12(b^2 − 1) = 12b^2 − 12.Step 9: Rearranging: 12b^2 − 70b − 12 = 0.Step 10: Divide the whole equation by 2 to simplify: 6b^2 − 35b − 6 = 0.Step 11: Solve the quadratic 6b^2 − 35b − 6 = 0 (for example by factoring or quadratic formula). The positive root is b = 6.Step 12: Therefore, the speed of the motor boat in still water is 6 km/h.

Verification / Alternative check:
With b = 6 km/h, upstream speed = 6 − 1 = 5 km/h and downstream speed = 6 + 1 = 7 km/h. Upstream time for 35 km = 35 / 5 = 7 hours. Downstream time for 35 km = 35 / 7 = 5 hours. Total time = 7 + 5 = 12 hours, which matches the given total time, confirming that b = 6 km/h is correct.

Why Other Options Are Wrong:
Speeds like 8 or 7.5 km/h do not give a total time of 12 hours when substituted into the equation. A speed of 5.5 km/h yields non-integer and incorrect total time compared to 12 hours. A speed of 9 km/h would make the boat too fast and total time too short. Only 6 km/h leads to a perfect match with the given total round-trip time.

Common Pitfalls:

• Using 35 km as the total distance instead of 70 km (35 upstream + 35 downstream).
• Writing upstream speed as b + 1 instead of b − 1.
• Algebraic errors when combining fractions or when solving the quadratic equation.
• Trying to guess the answer without verifying that the total time is exactly 12 hours.

Final Answer:
The speed of the motor boat in still water is 6 km/h.